Polar domain walls are currently at the focus of intensive research owing to their unusual and highly localized functional properties, which bear great potential for technological applications. They can present unusual topological features, like swirling polar structures or defect lines. The prediction of possible non-Ising and chiral internal structures of polar domain walls has been a particularly important development in this topic over the past years. This Tutorial highlights the capabilities of non-linear optics to probe these newly discovered aspects in polar non-Ising type domain walls through the second-harmonic generation (SHG) process. Fundamental symmetry properties of domain walls are presented in the context of recent advances on chiral and abnormal polar structures. We introduce the basics of the SHG and its ability to probe the symmetry down to the nanoscale, and we explain how to obtain insight into the non-Ising character of polar domain walls by combining the SHG polarimetry analysis with modeling.

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The wealth of properties in functional materials at the nanoscale has attracted tremendous interest over the last decades, spurring the development of ever more precise and ingenious characterization techniques. In ferroelectrics, for instance, scanning probe microscopy based techniques have been used in conjunction with advanced optical methods to probe the structure and properties of nanoscale domain walls, revealing complex behaviours such as chirality, electronic conduction or localised modulation of mechanical response. However, due to the different nature of the characterization methods, only limited and indirect correlation has been achieved between them, even when the same spatial areas were probed. Here, we propose a fast and unbiased analysis method for heterogeneous spatial data sets, enabling quantitative correlative multi-technique studies of functional materials. The method, based on a combination of data stacking, distortion correction, and machine learning, enables a precise mesoscale analysis. When applied to a data set containing scanning probe microscopy piezoresponse and second harmonic generation polarimetry measurements, our workflow reveals behaviours that could not be seen by usual manual analysis, and the origin of which is only explainable by using the quantitative correlation between the two data sets.